ANALYTICAL MECHANICS .ocity along any direction equals the rate at which distance .s covered along that direction. The velocity and its components evidently fulfill the -elation_____ v=^vx*+vv*+v*. (Ill) iVhen, as in the case of Fig. 48, the particle moves in the ;y-plane; z = 0, therefore V= V^/ + Vy2. Che direction of v, in this case, is given by tan0= v, (in') (IV) yhere 6 is the angle v makes with the ar-uxLs. ILLUSTRATIVE EXAMPLR Find the path, the velocity, and the components of the velocity of a •article which moves so that its position at any instant, is given by the blowing equations: x = at, (a) y = — Jf/r2. (1>) ]liminating t between (a) and (b), we obtain ^ 2(^ * ~ Ty/; )r the equation of the path, therefore the path in a parabola, Fig. -I!). To find the component-velocities we differentiate (a) and (10 with aspect to the time. This gives V x = a, _^ y = —gt. DISCUSSION. -— The horizontal cornpo-mt of the velocity is directed to the right id is constant, while the vertical corn-3nent is directed downwards and increases ; a constant rate. We will see later that these equations present the motion of a body which is projected horizontally from an evated position. I''KI. M.